= Cubic-matter no-scale vacuum with an exponential dilaton
{title2=$V_{\min}=0,\quad F^T\ne0\text{ on the nontrivial finite branch}$}
For $W=C^3+A+b$, $A=ae^{-\alpha S}$, the potential is $V=|C^3+b+(1+\alpha s)A|^2/(st^3)+3|C|^4/(st^2)$. A finite zero-energy vacuum requires $C=0$ and $b+(1+\alpha s)A=0$. With $a,b\ne0$, such a solution exists exactly when $0<|b/a|\le2e^{-1/2}$: the magnitude equation is $|b/a|=(1+x)e^{-x/2}$, $x=\alpha s>0$, whose maximum occurs at $x=1$. At that vacuum $F^S=F^C=0$ but $F^T=e^{K/2}t\bar W\ne0$. Both components of $T$ are flat, while $C$ is quartically lifted. If no root exists, the positive potential falls toward zero as $t\to\infty$ and has no finite minimum. If $a=b=0$, the $C=0$ family is supersymmetric instead.
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